Effect of Orientation and Temperature on Edge Crack
Propagation of Bcc Iron under Cyclic Loading
L Ma
*
, C S Li and J R Guo
College of Physics and Electronic Science, Hunan University of Arts and Science,
Changde 41500 China
Abstract. The fat igue crack propagation behaviour with pre-existing edge crack was
investigated in iron single crystal by mo lecular dynamics simulat ion. The results showed that
the deformat ion characteristics of crack tip we re influenced by the orientation of init ial crack.
The (010) [001] crack presented ductile fracture and the slip bands of crack tip were
{110}<111> systems at room temperature. Nevertheless, the (
1
10) [110] crack presented
brittle rupture and the slip bands of crack tip were (011) [1
1
1] system. In (111) [11
2
] crack,
the main deformation of crack tip was vacancies and dislocations at room temperature. The
influence of temperature on the propagation of crack was also discussed in all crack models,
the results revealed that the crack growth rate and the deformat ion mechanisms of crack tip
changed as the temperature rose, and the threshold values of stress intensity factor decreased
with the increasing temperature.
1. Introduction
The behaviour of fatigue crack propagation has a significant influence on the fracture properties of
metallic materials. The studies of experimental and theoretical indicate that the phenomenon of
brittle-to-ductile (BDT) transition is the main deformation characteristic during crack propagation in
metallic materials [1-7]. The molecular dynamics (MD) simulations, which can provide the
microstructure evolution of the materials, is a valuable tool and used to simulate the behaviour of
crack propagation. The study results show that the phase transition and recrystallization are found at
crack tip in body-centered cubic (bcc) iron at a higher loading level for {110} <110> and {111} <110>
cracks [8-9]. Uhnakova et al [10-11] also predicted the emitting of dislocations at crack tip and
analyzed the shielding or anti-shielding effects of dislocations and twins using MD simulations in bcc
iron. In addition, the interaction between crack tip and pre-existing dislocations was also investigated,
and indicated that it could cause the generation and motion of new dislocations at crack tip [12].
Meanwhile, much attention had been devoted to the investigation of crack propagation behaviour
under cyclic loading [13-15].
In the studies of fatigue crack propagation behaviour, many researchers have been achieved
significant results. It is concluded that the persistent slip bands (PSBs), voids and twins are observed
at crack tip in the process of fatigue crack propagation for face-center cubic (fcc) metals [16-17].
Nishimura and Miyazaki [18] also investigated the interaction between the crack and two tilt grain
boundaries using MD simulations and indicated that the phase transition and vacancies caused by the
emission and absorption of dislocations were observed around the crack tip. Tang et al [19] analyzed
20
Ma, L., Li, C. and Guo, J.
Effect of Orientation and Temperature on Edge Crack Propagation of Bcc Iron under Cyclic Loading.
In Proceedings of the International Workshop on Materials, Chemistry and Engineering (IWMCE 2018), pages 20-30
ISBN: 978-989-758-346-9
Copyright © 2018 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

the fatigue crack propagation behaviour in magnesium single crystal under cyclic loading and
revealed that the fatigue crack growth rate and the crack path varied with the crystallographic
orientation of initial crack, and the strain rate and temperature had an important influence on crack
growth rate. Uhnakova et al [20] investigated the ductile-brittle behaviour of an edge (
1
10) [110]
crack in bcc iron at room temperature using MD simulations, the results revealed that the crack
emitted dislocations in the inclined <111>{112} slip system under cyclic loading. Prahl et al [21]
analyzed the behaviour of ductile-brittle at the (110)[001] crack in bcc iron crystal loaded
monotonically in mode I, the fracture experiment showed that the cracks were deviated along {100}
planes and the fracture was accompanied by dislocation slip and twining, and also indicated that 3D
simulations using molecular dynamics simulations contributed to the understanding of crack
propagation. Anna et al [22] presented 3D molecular simulations of edge crack (001) [110] in bcc
iron under stress cyclic loading in mode I and II at room temperature, and showed that the crack (001)
[110] loaded cyclically in mode I generated dislocations in oblique slip systems <11
1
> {011}, further
oblique twins in <111> {11
2
} and inclined twins in slip systems <1
1
1> {
1
12} at higher loading.
This paper was devoted to investigating the edge crack propagation behaviour in bcc iron single
crystal under cyclic loading using MD simulations. The models of (010)[001] crack, (
1
10)[110]
crack and (111)[11
2
] crack were used to study the behaviour of fatigue crack propagation. The
deformation characteristics of crack tip were displayed. The effects of temperature on fatigue crack
propagation were discussed for all the crack models, and the crack growth rate and stress intensity
factor threshold were given under various temperatures.
2. Atomistic model and simulation method
The specimens of initial crack models were sketched in Figure 1(a). Pre-existing edge crack was built
by removing atoms in iron single crystal. The crack surface was perpendicular in the y- direction and
the crack front was oriented along the z- direction. The (010) [001] crack, (
1
10)[110] crack and
(111)[11
2
] crack were created, and the loading directions were [010], [
1
10] and [111], respectively.
For the crack models, the numbers in parentheses represented initial crack plane and those in square
brackets represented the front of initial crack. The periodic boundary was applied in z- direction and
other directions were applied to non-periodic boundary. About five atomic layers lied in upper and
lower boundaries were fixed and the cyclic loading was applied in y-direction. Al the crack models
were built with approximate dimensions of 57.2nm × 57.2nm × 5.72nm in x-, y- and z- direction, the
total number of atoms was 1600000. The initial crack length was 5.72 nm, as 10 percent of the length
of simulation boxes. Figure 1(b) showed the mode of strain cyclic loading, the initial strain amplitude
of cyclic loading was 0.01 and the strain ratio was at the level of
5.0/
maxmin
  R
. Before cyclic
loading, all the crack models were relaxation at various simulated conditions and reached equilibrium.
The numbers of cyclic loading ranged from 1 to 15 for all crack models.
In the MD simulations, the interactions between atoms were described by a modified analytic
embedded atom method (MAEAM), which had been applied successfully for the studies of metallic
microstructure and melting simulations [23-27]. The MAEAM is a type of EAM potential and its
formalisms is
)()()(
2
1
ii
ij
iji
PMFrE 



(1)
where
)(
ij
r 
is the pairwise interaction,
)(
i
F 
present the embedding energy,
)(
i
PM
is an extra
energy-modified term that represents the contribution of non-spherical symmetry coming from each
neighbor atom
j
with respect to the spherically symmetric atomic electron density.
Effect of Orientation and Temperature on Edge Crack Propagation of Bcc Iron under Cyclic Loading
21

The microstructure evolution was analyzed with common neighbor analysis (CNA) proposed by
Honeycutt and Andersen [28], which provided the details of evolution process of fatigue crack
propagation. The MD code of LAMMPS was used to study the fatigue crack propagation in this
paper [29].
(a) (b)
Figure 1. (a) Geometry of the atomistic model, (b) the mode of cyclic loading.
3. Results and discussion
3.1. Fatigue crack growth in iron single crystal
In this section, we investigated the behaviour of fatigue crack propagation in iron single crystal at
room temperature. The stress intensity factor (
I
K
), which described the blunting phenomenon at
crack tip, was calculated by Griffith level
2/1
0
)( lFKK
AIGI
 
,
A

was the applied stress,
0
l
was the
crack length,
16.1
I
F
was the boundary correction factor [4, 20]. For fatigue crack propagation, the
stress intensity factor range (
K
) was calculated by
minmax
KKK 
, the values of the maximum
(
max

) and minimum (
min

) stresses were used to calculate
max
K
and
min
K
, The
K
was between the
threshold of stress intensity factor range (
th
K
) and the crack fracture toughness (
Ic
K
). As the value
of
K
were larger that
th
K
, the crack began to cleavage. Figure 2 showed fatigue crack propagation
behaviour analyzed by CNA for (010)[001] crack in iron single crystal at room temperature. In
Figure 2, the blue color denoted the perfect iron lattice (with coordination number KNT=14), the
green color represented the slip bands (KNT=16). Observed from the (001) plane of model, up to
cycle 5, the shearing slip bands were observed around the crack tip and the slip systems were along
<111>{110} slip systems as the result of local stress concentration at crack tip. The Schmid factor
described the slip systems was given and the value was 0.41, which was consistent with the inclined
systems {110} simulated by Alena [20]. As the cyclic loading increased, the plastic deformation
zones were expansion, and blunting atom occurred at crack tip due to stress concentration at cycle 6,
and we obtained
043.2
G
K
MPa m
1/2
and
Ic
K
th
K
= 0.299 MPa m
1/2
at the moment. In the case of
cycle 7, the crack presented ductile fracture and propagated in the form of sharp crack along with [11
1
] direction, and the slip bands appeared at sharp crack tip. Figure 3(a) and Figure 3(b) showed the
detail and 3D visualization of slip bands at crack tip at cycle 7. From Figure 3, it was clear that the
complete dislocation obtained from crack tip due to influence of the periodic boundary conditions.
IWMCE 2018 - International Workshop on Materials, Chemistry and Engineering
22

The shearing slip bands caused by the motion of dislocations significantly decreased the crack
growth rate by reducing stress concentration at crack tip. At cycle 8, the crack propagated continually
in accompany with the slip bands of crack tip expanding, the blunting and voids were the main
deformation mechanisms of crack tip caused by the stress concentration and induced the fatigue
crack propagation.
Cycle 5 Cycle 6 Cycle 7 Cycle 8
Figure 2. The microstructure evolution of crack tip analyzed by CNA for (010)[001] crack under
cyclic loading in iron single crystal.
(a) (b)
Figure 3. (a) The generation of slip band (green atoms in front of the crack) at crack tip in surface
layer of (001) of (010)[001] crack at cycle 7, (b) 3D visualization of slip bands around the crack tip.
Figure 4 showed the deformation characteristics of crack tip at different crack models as the
fatigue crack propagated. The blue color represented the iron atoms and the green color represented
the slip bands. For (
1
10)[110] crack and (111)[11
2
] crack, we obtained
054.2
G
K
MPa m
1/2
,
th
K
= 0.216 MPa m
1/2
and
143.2
G
K
MPa m
1/2
,
th
K
=0.116 MPa m
1/2
, respectively. These values were
lower comparing with (010)[001] crack. Figure 4(a) showed the microstructure evolution of crack tip
analyzed by CNA for (
1
10) [110] crack under strain cyclic loading in iron single crystal. The results
indicated that the main deformation mechanisms were shearing slip bands at crack tip, and the slip
system were (011)[1
1
1]. The crack presented brittle rupture and the path of crack propagation was
along [
1
10] direction. Figure 5(a1) and Figure 5(a2) presented the generation of slip bands at crack
tip in surface layer (110) of (
1
10)[110] crack at cycle 6. In comparison with the stress cyclic loading
in the (
1
10)[110] crack, the oblique slip changed and the {112}<111> slip systems [20] were not
observed. Figure 4(b) presented the behaviour of fatigue crack propagation for (111) [11
2
] crack.
The results showed that the formation of micro-crack at crack tip induced the crack propagation
along [
1
10] direction in accompany with blunting phenomenon caused by stress concentration, and
the dislocations were the dominant deformation around the crack tip. Figure 5(b1) and Figure 5(b2)
Effect of Orientation and Temperature on Edge Crack Propagation of Bcc Iron under Cyclic Loading
23

showed the deformation mechanisms of crack tip in surface layer (11
2
) of (111) [11
2
] crack at
cycle 4. The vacancies were also observed at crack tip. As seen from above results, it was shown that
the front of initial crack had an important influence on the deformation mechanisms of crack tip for
iron single crystal under cyclic loading at room temperature.
(a) Cycle 5 (a) Cycle 6 (b) Cycle 4 (b) Cycle 5
Figure 4. The microstructure evolution of crack tip analyzed by CNA for (a) (
1
10)[110] crack and (b)
(111)[11
2
] crack under cyclic loading in iron single crystal.
(a1) (a2)
(b1) (b2)
Figure 5. (a1) The generation of slip bands at crack tip in surface layer (110) of (
1
10)[110] crack at
cycle 6, (a2) 3D visualization of dislocations emission around the crack tip in (
1
10)[110] crack
(green atoms in front of the crack), (b1) The deformation mechanisms of crack tip in surface layer
(11
2
) of (111)[11
2
] crack at cycle 4, (b2) 3D visualization of deformation mechanism around the
crack tip in (111)[11
2
] crack (green atoms in front of the crack).
3.2. Crack growth rate
In this section, the variation of crack length with number of cycles for various crack models was
calculated by tracking the position of crack tip as shown in Figure 6. The simulation results showed
that the length of (
1
10)[110] crack grew faster comparing with other crack models. The average
crack growth rates of (010)[001] crack, (
1
10)[110] crack and (111)[11
2
] crack by fitting the
variation of crack length were 2.46×10
-9
m/cycle, 6.59×10
-9
m/cycle and 2.19×10
-9
m/cycle,
respectively. So the crack growth rate of (
1
10) [110] crack was maximum, and the crack growth rate
of (111) [11
2
] crack was minimum. The reasons resulting in the differences of crack growth rate for
Blunting
Micro-crack
Dislocation
s
IWMCE 2018 - International Workshop on Materials, Chemistry and Engineering
24

the cracks of various orientations were the differences of deformation mechanisms of crack tip.
Besides, the interaction between the crack growth rate (
dNda/
) and the stress intensity factor range
(
K
) was calculated from the known relations
n
thth
KKdNdadNda )/()/(/ 
, where
a
is the
crack half length,
N
is the number of cycles, “th” was the threshold, “n” was Paris exponent. The
value of Paris exponent was obtained by the results of atomistic simulation of fatigue behaviour in
bcc iron [10]. Figure 7 presented the crack growth rate with the change of stress intensity factor range
in different crack models. The results showed that the values of
dNda/
ranged from 10
-11
to 10
-9
m/cycle. The
K
of the crack models ranged from 0.105 to 0.548
mMPa
, these values were lower
comparing with the center crack propagation in iron single crystal [30]. At the same time, the crack
growth rate of all the crack models increased as the increase of
K
, and the (
1
10)[110] crack had the
maximum rate of crack propagation comparing with other crack models.
0 2 4 6 8 10
0
200
400
600
crack length / 10
-10
m
cycle number
(010)[001]crack
(-110)[110]crack
(111)[11-2]crack
0.1 0.2 0.3 0.4 0.5 0.6 0.7
0
2
4
6
8
10
(010)[001]crack
(-110)[110]crack
(111)[11-2]crack
da/dN / 10
-10
m
Pa m
1/2
Figure 6. The variation of crack length was Figure 7. Crack growth rate ranged with the
stress calculated at crack the model of different intensity factor range for (010)[001] crack,
front under cyclic loading. (
1
10) [110] crack and (111) [11
2
] crack.
3.3. Temperature effect
In order to analyze the temperature effects on the behaviour of fatigue crack propagation, various
temperatures including 300K, 400K, 500K, and 600K were applied to all the crack models. The blue,
green and red colors represented bcc structure, fcc slip and hcp structure in Figure 8, Figure 9 and
Figure 10, respectively. Figure 8 presented the behaviour of fatigue crack propagation for (010)[001]
crack at various temperatures. It showed that the dominant deformation mechanisms were shearing
slip bands, blunting and linkage of voids at the tip of crack as the increasing temperature. The crack
presented ductile fracture as crack propagation at temperature of 300K. From 400K to 600K, the
crack appeared softening and the growth path of crack changed with the formation of sharp crack at
crack tip. Besides, the phase transition of local region was also observed in the front of crack from
400K to 600K. Figure 9 showed the propagation behaviour of crack for (
1
10) [110] crack at various
temperatures in single crystal iron. The results indicated that the deformation mechanism of crack tip
had changed at higher temperatures, and the voids were the dominate mechanism at crack tip at 400K.
From 500K to 600K, some microcracks appeared at the crack front and induced the structure failure.
For the (111)[11
2
] crack, temperature mainly affected the crack propagation path as shown in Figure
10. At room temperature the (111)[11
2
] crack advanced along [110] direction, nevertheless, the path
of crack propagation deviated from the [110] direction as the temperature rose. At high temperatures,
Effect of Orientation and Temperature on Edge Crack Propagation of Bcc Iron under Cyclic Loading
25

the sharp crack that formed along [111] direction at crack tip had grown larger and forced the path of
crack propagation to widen that resulting in the rate of crack propagate increasing for (111)[11
2
]
crack.
300K 400K 500k 600k
Figure 8. The deformation behaviour of crack tip analyzed by CNA for (010)[001] crack at various
temperatures in iron single crystal.
300K 400k 500K 600K
Figure 9. The deformation behaviour of crack tip analyzed by CNA for (
1
10) [110] crack at various
temperatures in iron single crystal.
300K 400K 500K 600K
Figure 10. The deformation behaviour of crack tip analyzed by CNA for (111)[11
2
] crack at various
temperatures in iron single crystal.
Figure 11 showed the variation of crack length for (010)[001] crack, (
1
10)[110] crack and
(111)[11
2
] crack at various temperatures with number of cycles in iron single crystal. In (010)[001]
crack, the results indicated that the crack growth rate was basically to assume decline with the
increasing temperature in addition to 500K. For the special temperature of 500K, the crack growth
rate increased comparing with others. The reason was that the formation of voids around the crack tip
led to the increasing of crack propagation rate. In (
1
10) [110] crack, the obtained from the variation
of crack length with the number of cycles at various temperatures showed that the fatigue crack
Sli p bands
Cracks
Crack propagation path
Voids
IWMCE 2018 - International Workshop on Materials, Chemistry and Engineering
26

growth rate decreased as the temperature rose, because the materials became softening with the
increasing temperature and effectively inhibited the fatigue crack propagation. However, the
(111)[11
2
] crack presented a different temperature effect, the fatigue crack growth rate increased
with the increasing temperature as showed in Figure 11(c).
5 6 7 8 9 10
0
100
200
300
crack length / 10
-10
m
cycle number
300K
400K
500K
600K
4 5 6 7 8
0
100
200
300
400
crack length / 10
-10
m
cycle number
300K
400K
500K
600K
(a) (b)
4 5 6 7 8
0
100
200
300
400
500
crack length / 10
-10
m
cycle number
300K
400K
500K
600K
(c)
Figure 11. The variation of crack length for (a): (010) [001] crack, (b): (
1
10) [110] crack and (c):
(111) [11
2
] crack at various temperatures under cyclic loading in iron single crystal.
300 400 500 600
0.0
0.1
0.2
0.3

th
/ MPa m
1/2
Temperature / K
(010)[001]crack
[-110][110]crack
(111)[11-2]crack
Figure 12. The variation of the stress intensity factor threshold (
th
K
) for (010) [001] crack, (
1
10)
[110] crack and (111)[11
2
] crack at various temperatures in iron single crystal.
The fracture toughness of crack tip under cyclic loading is described by the given value of the
stress intensity factor threshold (
th
K
) which was based on the Griffith fracture theory. As the stress
Effect of Orientation and Temperature on Edge Crack Propagation of Bcc Iron under Cyclic Loading
27

intensity factor
K
was below the value of
th
K
, the mechanical load can not lead to the crack
initiation and propagation. Figure 12 showed the variation of the threshold value of stress intensity
factor
th
K
at various temperatures for (010) [001] crack, (
1
10)[110] crack and (111)[11
2
] crack.
The simulation results presented that the
th
K
decreased with the increasing temperature for all the
crack models, which was caused by the various deformation mechanisms of crack tip. In (010)[001]
crack, as the temperature rose, the slip bands of crack tip increased as shown in Figure 8 so that the
stress concentration of crack tip was relaxed, and led to the
th
K
decreasing. For (
1
10)[110] crack
and (111)[11
2
] crack, a large number of voids and the formation of sharp crack destroyed the stress
concentration of crack tip at higher temperatures as shown in Figure 9 and Figure 10so that the
driving force of crack tip decreased, thus resulting in the decreasing of
th
K
. So the fracture mode of
material was the ductile fracture caused by crack propagation with the increasing temperatures, and
the high temperature was effective on the resistance of fatigue crack propagation.
4. Conclusions
The behaviour of fatigue crack propagation with pre-existing edge crack was investigated in iron
single crystal under strain cyclic loading using MD simulations. The deformation mechanisms of
crack tip were depended on the crack front. In (010)[001] crack, the crack presented ductile fracture
and the dominant mechanism were slip bands and the slip systems was <111>{110}. For (
1
10) [110]
crack, the crack propagation was brittle rupture and the slip bands was along (011) [1
1
1] system.
However, in (111) [11
2
] crack, the micro-crack occurred at crack tip and induced the crack
propagation, and the main deformation mechanisms of crack tip were vacancies and dislocations.
Meanwhile, the (
1
10)[110] crack had the maximum crack growth rate comparing with (010)[001]
crack and (111)[11
2
] crack.
The effects of temperature on fatigue crack propagation in iron single crystal were studied. For
(010)[001] crack, at high temperature the deformation mechanisms of crack tip were the linkage of
voids slip bands and phase transition that happened in front of crack tip. The crack growth rate
decreased with increasing temperature except for 500K which was caused by the formation of the
linkage of voids at crack tip. For (
1
10) [110] crack, as the temperature rose, voids were the
predominate deformation around the crack tip, new micro-cracks was formation at the crack front,
and the fatigue crack growth rate decreased. However, the fatigue crack growth rate of (111) [11
2
]
crack increased with the increasing temperature. Besides, the threshold of stress intensity factor of all
the crack models decreased with the increasing temperature.
Acknowledgements
This work is supported by the “16BSQD05 and the Key Youth Foundation of Hunan Provincial
Education Department (No.17B180) and the Jiangxi Provincial Natural Science Foundation of China
(No. 20171BAB216001) and Scientific Research Project of Jiangxi Provincial Education Department
(No. GJJ161242).
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